Affine Deligne-Lusztig varieties at infinite level

DOI10.1007/S00208-020-02092-4arXiv1811.11204WikidataQ123125210 ScholiaQ123125210MaRDI QIDQ6310315

Publication date: 27 November 2018

Abstract: We initiate the study of affine Deligne-Lusztig varieties with arbitrarily deep level structure for general reductive groups over local fields. We prove that for GLn and its inner forms, Lusztig's semi-infinite Deligne-Lusztig construction is isomorphic to an affine Deligne-Lusztig variety at infinite level. We prove that their homology groups give geometric realizations of the local Langlands and Jacquet--Langlands correspondences in the setting that the Weil parameter is induced from a character of an unramified field extension. In particular, we resolve Lusztig's 1979 conjecture in this setting for minimal admissible characters.

Mathematics Subject Classification ID

Étale and other Grothendieck topologies and (co)homologies (14F20) Varieties over finite and local fields (11G25) Linear algebraic groups over local fields and their integers (20G25)

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